Time-space Finite Element Adaptive AMG for Multi-term Time Fractional Advection Diffusion Equations

In this study we construct a time-space finite element (FE) scheme and furnish cost-efficient approximations for one-dimensional multi-term time fractional advection diffusion equations on a bounded domain $\Omega$. Firstly, a fully discrete scheme is obtained by the linear FE method in both temporal and spatial directions, and many characterizations on the resulting matrix are established. Secondly, the condition number estimation is proved, an adaptive algebraic multigrid (AMG) method is further developed to lessen computational cost and analyzed in the classical framework. Finally, some numerical experiments are implemented to reach the saturation error order in the $L^2(\Omega)$ norm sense, and present theoretical confirmations and predictable behaviors of the proposed algorithm.